Quadrilateral meshes for PSLGs

TL;DR

This paper proves that any planar straight line graph can be meshed with quadrilaterals having optimal angle bounds and complexity, improving mesh quality and demonstrating sharp bounds.

Contribution

It introduces a method to generate conforming quadrilateral meshes with optimal angle bounds and complexity for any planar straight line graph.

Findings

Quadrilateral meshes with $O(n^2)$ elements are possible for any PSLG.

All angles in the mesh can be bounded between 60° and 120°.

Most angles can be made nearly equal, around 90°.

Abstract

We prove that every planar straight line graph with $n$ vertices has a conforming quadrilateral mesh with $O(n^2)$ elements, all angles $\leq 120^\circ$ and all new angles $\geq 60^\circ$. Both the complexity and the angle bounds are sharp. Moreover, all but $O(n)$ of the angles may be taken in a smaller interval, say $[89^\circ, 91^\circ]$.